It is the time evolution of a quantity. Signal is a way of conveying information. Gestures, images, sound, all can be signals. Technically - a function of time, space, or another observation variable that conveys information Some examples of the signals are:
The price of a share in a publically listed company;
The level of water in a reservoir;
The speed of a car;
The temperature in a room;
The voltage driving the speaker in a mobile telephone;
many others . . .
In the case of voltage across a branch, the signal is expressed as a function of time v(t), whereas in the case of a charge distributed over a line, the signal is the charge density expressed as a function of space as k(x).
When the function depends on a single variable, the signal is said to be one dimensional. A speech signal is an example of a one-dimensional signal whose amplitude varies with time.
When the function depends on two or more variables, the signal is said to be multidimensional. An image is an example of a two-dimensional signal, with the horizontal and vertical coordinates of the image representing the two dimensions.
Deterministic and random:
A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time.
Signals can be deterministic in that their values can be exactly modeled as a completely specified function of time.
Random signals take on a random values at any given time instant.
(we will deal with deterministic signals only)
We will further distinguish three main forms of signals:
Continuous-Time/Analog Signal
Discrete-Time Signal
Digital Signal
Continuous-time (CT)/Analog Signal
A finite, real-valued, smooth function s(t) of a variable t which usually represents time. Both s and t in s(t) are continuous
Why real-valued?
Usually real-world phenomena are real-valued.
Why finite?
Real-world signals will generally be bounded in energy, simply because there is no infinite source of energy available to us.
Why smooth?
Real world signals never change abruptly/instananeously.
Discrete-time(DT) Signal
A discrete-time signal is a bounded, continuous-valued sequence s[n]. Alternately, it may be viewed as a continuous-valued function of a discrete index n. We often refer to the index n as time, since discrete-time signals are frequently obtained by taking snapshots of a continuous-time signal as shown below. More correctly, though, n is merely an index that represents sequentiality of the numbers in s[n].
If they DT signals are snapshots of real-world signals realness and finiteness apply.
Digital Signal (we will not deal with this signal)
We will work with digital signals but develop theory mainly around discrete-time signals. Digital computers deal with digital signals, rather than discrete-time signals. A digital signal is a sequence s[n], where index the values s[n] are not only finite, but can only take a finite set of values. For instance, in a digital signal, where the individual numbers s[n] are stored using 16 bits integers, s[n] can take one of only 216 values. In the digital valued series s[n] the values s can only take a fixed set of values. Digital signals are discrete-time signals obtained after "digitalization." Digital signals too are usually obtained by taking measurements from real-world phenomena. However, unlike the accepted norm for analog signals, digital signals may take complex values.
And also go through following pages for more reading on signal and system.
Periodic and aperiodic signal: A periodic signal is one for which x(t+To) = x(t) where To is termed the period. The smallest value of To for which the above equation holds is called the fundamental period. ( It should be clear that if To is the fundamental period, then any integer multiple of To is also a period.)
Energy and Power Signals:This is another classification of signals. I recommend you to see the following video to understand this.
Time Scaling:
The compression or expression of a signal in time is known as the time scaling. If x(t) is the original signal then x(at) represents its time scaled version. Where a is constant.
If a> 1 then x(at) will be a compressed version of x(t) and if a< 1 then it will be a expanded version of x(t). Example: Let x(t) = u(t) – u(t – 2). Sketch y(t) = x(2t)
Multiplying Signals (Amplitude Modulation) Audio signals are great for short distance communication; when you speak, someone across the room hears you. On the other hand, radio frequencies are very good at propagating long distances. For instance, if a 100 volt, 1 MHz sine wave is fed into an antenna, the resulting radio wave can be detected in the next room, the next country, and even on the next planet. Modulation is the process of merging two signals to form a third signal with desirable characteristics of both. This always involves nonlinear processes such as multiplication; you can't just add the two signals together. In radio communication, modulation results in radio signals that can propagate long distances and carry along audio or other information.
Radio communication is an extremely well developed discipline, and many modulation schemes have been developed. One of the simplest is called amplitude modulation. Figure 10-14 shows an example of how amplitude modulation appears in both the time and frequency domains. Continuous signals will be used in this example, since modulation is usually carried out in analog electronics. However, the whole procedure could be carried out in discrete form if needed (the shape of the future!).
Figure (a) shows an audio signal with a DC bias such that the signal always has a positive value. Figure (b) shows that its frequency spectrum is composed of frequencies from 300 Hz to 3 kHz, the range needed for voice communication, plus a spike for the DC component. All other frequencies have been removed by analog filtering. Figures (c) and (d) show the carrier wave, a pure sinusoid of much higher frequency than the audio signal. In the time domain, amplitude modulation consists of multiplying the audio signal by the carrier wave. As shown in (e), this results in an oscillatory waveform that has an instantaneous amplitude proportional to the original audio signal. In the jargon of the field, the envelope of the carrier wave is equal to the modulating signal. This signal can be routed to an antenna, converted into a radio wave, and then detected by a receiving antenna. This results in a signal identical to (e) being generated in the radio receiver's electronics. A detector or demodulator circuit is then used to convert the waveform in (e) back into the waveform in (a).
Since the time domain signals are multiplied, the corresponding frequency spectra are convolved. That is, (f) is found by convolving (b) & (d). Since the spectrum of the carrier is a shifted delta function, the spectrum of the modulated signal is equal to the audio spectrum shifted to the frequency of the carrier. This results in a modulated spectrum composed of three components: a carrier wave, an upper sideband, and a lower sideband
.
Time Invariant System : And some more problems on Time Invariant property of the system And one more problem
What is a Signal?
It is the time evolution of a quantity. Signal is a way of conveying information. Gestures, images, sound, all can be signals.
Technically - a function of time, space, or another observation variable that conveys information
Some examples of the signals are:
In the case of voltage across a branch, the signal is expressed as a function of time v(t), whereas in the case of a charge distributed over a line, the signal is the charge density expressed as a function of space as k(x).
When the function depends on a single variable, the signal is said to be one dimensional. A speech signal is an example of a one-dimensional signal whose amplitude varies with time.
When the function depends on two or more variables, the signal is said to be multidimensional. An image is an example of a two-dimensional signal, with the horizontal and vertical coordinates of the image representing the two dimensions.
Deterministic and random:
A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time.Signals can be deterministic in that their values can be exactly modeled as a completely specified function of time.
Random signals take on a random values at any given time instant.
(we will deal with deterministic signals only)
We will further distinguish three main forms of signals:
Continuous-time (CT)/Analog Signal
A finite, real-valued, smooth function s(t) of a variable t which usually represents time. Both s and t in s(t) are continuous
Why real-valued?
- Usually real-world phenomena are real-valued.
Why finite?- Real-world signals will generally be bounded in energy, simply because there is no infinite source of energy available to us.
Why smooth?Discrete-time(DT) Signal
A discrete-time signal is a bounded, continuous-valued sequence s[n]. Alternately, it may be viewed as a continuous-valued function of a discrete index n. We often refer to the index n as time, since discrete-time signals are frequently obtained by taking snapshots of a continuous-time signal as shown below. More correctly, though, n is merely an index that represents sequentiality of the numbers in s[n].
If they DT signals are snapshots of real-world signals realness and finiteness apply.
Digital Signal (we will not deal with this signal)
We will work with digital signals but develop theory mainly around discrete-time signals.Digital computers deal with digital signals, rather than discrete-time signals. A digital signal is a sequence s[n], where index the values s[n] are not only finite, but can only take a finite set of values. For instance, in a digital signal, where the individual numbers s[n] are stored using 16 bits integers, s[n] can take one of only 216 values.
In the digital valued series s[n] the values s can only take a fixed set of values.
Digital signals are discrete-time signals obtained after "digitalization." Digital signals too are usually obtained by taking measurements from real-world phenomena. However, unlike the accepted norm for analog signals, digital signals may take complex values.
And also go through following pages for more reading on signal and system.
Periodic and aperiodic signal:A periodic signal is one for which
x(t+To) = x(t)
where To is termed the period. The smallest value of To for which the above equation holds is called the fundamental period. ( It should be clear that if To is the fundamental period, then any integer multiple of To is also a period.)
Energy and Power Signals:This is another classification of signals. I recommend you to see the following video to understand this.Time Scaling:
The compression or expression of a signal in time is known as the time scaling. If x(t) is the original signal then x(at) represents its time scaled version. Where a is constant.If a> 1 then x(at) will be a compressed version of x(t) and if a< 1 then it will be a expanded version of x(t).
Example: Let x(t) = u(t) – u(t – 2). Sketch y(t) = x(2t)
(ref http://www.myclassbook.org/basic-operations-on-continuous-time-signal/)
Try to Sketch for y(t) = x(0.5t)
Multiplying Signals (Amplitude Modulation)
Audio signals are great for short distance communication; when you speak, someone across the room hears you. On the other hand, radio frequencies are very good at propagating long distances. For instance, if a 100 volt, 1 MHz sine wave is fed into an antenna, the resulting radio wave can be detected in the next room, the next country, and even on the next planet. Modulation is the process of merging two signals to form a third signal with desirable characteristics of both. This always involves nonlinear processes such as multiplication; you can't just add the two signals together. In radio communication, modulation results in radio signals that can propagate long distances and carry along audio or other information.
Radio communication is an extremely well developed discipline, and many modulation schemes have been developed. One of the simplest is called amplitude modulation. Figure 10-14 shows an example of how amplitude modulation appears in both the time and frequency domains. Continuous signals will be used in this example, since modulation is usually carried out in analog electronics. However, the whole procedure could be carried out in discrete form if needed (the shape of the future!).
Figure (a) shows an audio signal with a DC bias such that the signal always has a positive value. Figure (b) shows that its frequency spectrum is composed of frequencies from 300 Hz to 3 kHz, the range needed for voice communication, plus a spike for the DC component. All other frequencies have been removed by analog filtering. Figures (c) and (d) show the carrier wave, a pure sinusoid of much higher frequency than the audio signal. In the time domain, amplitude modulation consists of multiplying the audio signal by the carrier wave. As shown in (e), this results in an oscillatory waveform that has an instantaneous amplitude proportional to the original audio signal. In the jargon of the field, the envelope of the carrier wave is equal to the modulating signal. This signal can be routed to an antenna, converted into a radio wave, and then detected by a receiving antenna. This results in a signal identical to (e) being generated in the radio receiver's electronics. A detector or demodulator circuit is then used to convert the waveform in (e) back into the waveform in (a).
Since the time domain signals are multiplied, the corresponding frequency spectra are convolved. That is, (f) is found by convolving (b) & (d). Since the spectrum of the carrier is a shifted delta function, the spectrum of the modulated signal is equal to the audio spectrum shifted to the frequency of the carrier. This results in a modulated spectrum composed of three components: a carrier wave, an upper sideband, and a lower sideband
.
Time Invariant System :
And some more problems on Time Invariant property of the system
And one more problem